135:
692:
425:
798:
1542:
237:
155:
Step 2. Once solved, retain these known short rates, and proceed to the next time-step (i.e. input spot-rate), "growing" the tree until it incorporates the full input yield-curve.
1463:
2077:
514:
484:
140:
discount recursively through the tree using the rate at each node, i.e. via "backwards induction", from the time-step in question to the first node in the tree (i.e. i=0);
614:
1901:
583:
544:
2504:
1497:
934:
454:
750:
2034:
2014:
277:-prices for each component caplet); see aside. Using the calibrated lattice one can then value a variety of more complex interest-rate sensitive securities and
2418:
1451:
2335:
2345:
2019:
1457:
986:
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55:
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960:
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285:
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294:
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Although initially developed for a lattice-based environment, the model has been shown to imply the following continuous
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228:
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factor—the short rate—determines the future evolution of all interest rates. It was the first model to combine the
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24:
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1969:
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993:
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1984:
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1891:
1050:
551:
262:
163:
2719:
2559:
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2049:
1959:
1849:
1142:
718:
491:
461:
1030:
2689:
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2309:
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2127:
1964:
1629:
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918:
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702:
598:
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1808:
1701:
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1671:
1527:
1522:
1425:
1351:
1203:
1125:
781:
713:—are very easily applied to the calibration. Relatedly, the model was originally described in
557:
270:
266:
44:
find all other rates in the time-step, where these are linked to the node immediately above (r
2375:
2112:
521:
2729:
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1793:
1721:
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710:
232:
in 1990. A personal account of the development of the model is provided in
Emanuel Derman's
175:
144:
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1952:
1896:
1879:
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1741:
438:
799:"Society of Actuaries Professional Actuarial Specialty Guide Asset-Liability Management"
137:(this node-spacing being consistent with p = 50%; Δt being the length of the time-step);
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1996:
1991:
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1711:
1547:
1346:
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1238:
1168:
1094:
956:
935:"A One-Factor Model of Interest Rates and Its Application to Treasury Bond Options"
875:
786:
254:
2239:
751:"Impact of Different Interest Rate Models on Bond Value Measures, G, Buetow et al"
1012:
2704:
2223:
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2006:
1947:
1942:
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1666:
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1073:
992:. Technical Note No. 23, Options, Futures, and Other Derivatives. Archived from
258:
179:
257:
the model parameters to fit both the current term structure of interest rates (
2714:
2254:
2198:
2082:
195:
855:
2208:
1089:
714:
183:
34:
30:
143:
repeat until the discounted value at the first node in the tree equals the
1356:
1311:
274:
1036:
830:
2035:
Generalized autoregressive conditional heteroskedasticity (GARCH) model
1475:
893:"One on One Interview with Emanuel Derman (Financial Engineering News)"
233:
961:"Calibrating the Black–Derman–Toy model: some theoretical results"
697:
One reason that the model remains popular, is that the "standard"
41:
adjust the rate at the top-most node at the current time-step, i;
880:
Calibrating the Black–Derman-Toy model: some theoretical results
130:{\displaystyle \ln(r_{u}/r_{d})/2=\sigma _{i}{\sqrt {\Delta t}}}
1479:
1046:
1042:
1031:
R function for computing the Black–Derman–Toy short rate tree
222:, and Bill Toy. It was first developed for in-house use by
2015:
Autoregressive conditional heteroskedasticity (ARCH) model
1543:
Independent and identically distributed random variables
933:
Black, F.; Derman, E.; Toy, W. (January–February 1990).
827:"My Life as a Quant: Reflections on Physics and Finance"
192:
Lattice model (finance) § Interest rate derivatives
687:{\displaystyle d\ln(r)=\theta _{t}\,dt+\sigma \,dW_{t}}
595:
For constant (time independent) short rate volatility,
2020:
Autoregressive integrated moving average (ARIMA) model
1018:. Seminar Financial Engineering, University of Vienna.
15:
1464:
Securities
Industry and Financial Markets Association
625:
601:
560:
524:
494:
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441:
297:
58:
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2005:
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1365:
1267:
1161:
1103:
1013:"Implementation of the Black, Derman and Toy Model"
420:{\displaystyle d\ln(r)=\leftdt+\sigma _{t}\,dW_{t}}
686:
608:
577:
538:
508:
478:
448:
419:
129:
1902:Stochastic chains with memory of variable length
486:= value of the underlying asset at option expiry
882:, Applied Mathematical Finance 8, 27– 48 (2001)
194:. It is a one-factor model; that is, a single
1491:
1058:
8:
2030:Autoregressive–moving-average (ARMA) model
1498:
1484:
1476:
1452:Commercial Mortgage Securities Association
1065:
1051:
1043:
678:
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117:
111:
96:
87:
78:
72:
57:
1458:International Capital Market Association
878:, Ken Seng Tan and Weidong Tian (2001).
745:
743:
456:= the instantaneous short rate at time t
1037:Excel BDT calculator and tree generator
739:
2336:Doob's martingale convergence theorems
226:in the 1980s and was published in the
20:Short-rate tree calibration under BDT:
2088:Constant elasticity of variance (CEV)
2078:Chan–Karolyi–Longstaff–Sanders (CKLS)
926:Mathematica in Education and Research
202:behaviour of the short rate with the
7:
1388:Commercial mortgage-backed security
2575:Skorokhod's representation theorem
2356:Law of large numbers (weak/strong)
1383:Collateralized mortgage obligation
987:"The Black, Derman, and Toy Model"
119:
14:
2545:Martingale representation theorem
917:Benninga, S.; Wiener, Z. (1998).
2590:Stochastic differential equation
2480:Doob's optional stopping theorem
2475:Doob–Meyer decomposition theorem
919:"Binomial Term Structure Models"
286:stochastic differential equation
52:being the node in question) via
2460:Convergence of random variables
2346:Fisher–Tippett–Gnedenko theorem
516:= instant short rate volatility
2058:Binomial options pricing model
1378:Collateralized debt obligation
1254:Reverse convertible securities
641:
635:
376:
370:
313:
307:
206:, and is still widely used.
93:
65:
1:
2525:Kolmogorov continuity theorem
2361:Law of the iterated logarithm
509:{\displaystyle \sigma _{t}\,}
479:{\displaystyle \theta _{t}\,}
2530:Kolmogorov extension theorem
2209:Generalized queueing network
1717:Interacting particle systems
1011:Klose, C.; Li C. Y. (2003).
968:Applied Mathematical Finance
959:; Tan, K.; Tian, W. (2001).
214:The model was introduced by
1662:Continuous-time random walk
1194:Contingent convertible bond
147:corresponding to the given
2800:
2670:Extreme value theory (EVT)
2470:Doob decomposition theorem
1762:Ornstein–Uhlenbeck process
1533:Chinese restaurant process
1234:Inverse floating rate note
970:: 8, 27–48. Archived from
942:Financial Analysts Journal
229:Financial Analysts Journal
2738:
2550:Optional stopping theorem
2351:Large deviation principle
2103:Heath–Jarrow–Morton (HJM)
2040:Moving-average (MA) model
2025:Autoregressive (AR) model
1850:Hidden Markov model (HMM)
1784:Schramm–Loewner evolution
1080:
609:{\displaystyle \sigma \,}
279:interest rate derivatives
188:interest rate derivatives
2465:Doléans-Dade exponential
2295:Progressively measurable
2093:Cox–Ingersoll–Ross (CIR)
1393:Mortgage-backed security
1162:Types of bonds by payout
1104:Types of bonds by issuer
852:"Black-Derman-Toy (BDT)"
717:language, and not using
578:{\displaystyle dW_{t}\,}
27:of an up move, p, to 50%
25:risk-neutral probability
2685:Mathematical statistics
2675:Large deviations theory
2505:Infinitesimal generator
2366:Maximal ergodic theorem
2285:Piecewise-deterministic
1887:Random dynamical system
1752:Markov additive process
944:: 24–32. Archived from
699:Root-finding algorithms
539:{\displaystyle W_{t}\,}
204:log-normal distribution
178:used in the pricing of
151:for the i-th time-step.
29:Step 1. For each input
2520:Karhunen–Loève theorem
2455:Cameron–Martin formula
2419:Burkholder–Davis–Gundy
1814:Variance gamma process
1327:Option-adjusted spread
1229:Inflation-indexed bond
688:
610:
579:
540:
510:
480:
450:
421:
168:Black–Derman–Toy model
131:
2769:Fixed income analysis
2650:Actuarial mathematics
2612:Uniform integrability
2607:Stratonovich integral
2535:Lévy–Prokhorov metric
2439:Marcinkiewicz–Zygmund
2326:Central limit theorem
1928:Gaussian random field
1757:McKean–Vlasov process
1677:Dyson Brownian motion
1538:Galton–Watson process
1373:Asset-backed security
1337:Weighted-average life
1174:Auction rate security
782:Fixed Income Analysis
689:
611:
580:
554:probability measure;
541:
511:
481:
451:
422:
132:
2725:Time series analysis
2680:Mathematical finance
2565:Reflection principle
1892:Regenerative process
1692:Fleming–Viot process
1507:Stochastic processes
1366:Securitized products
623:
599:
558:
522:
492:
462:
439:
295:
263:volatility structure
164:mathematical finance
56:
2720:Stochastic analysis
2560:Quadratic variation
2555:Prokhorov's theorem
2490:Feynman–Kac formula
1960:Markov random field
1608:Birth–death process
1143:Infrastructure bond
719:stochastic calculus
449:{\displaystyle r\,}
351:
249:Under BDT, using a
2690:Probability theory
2570:Skorokhod integral
2540:Malliavin calculus
2123:Korn-Kreer-Lenssen
2007:Time series models
1970:Pitman–Yor process
1219:Floating rate note
684:
606:
575:
536:
506:
476:
446:
417:
339:
267:interest rate caps
238:My Life as a Quant
149:spot interest rate
127:
2784:Options (finance)
2774:Short-rate models
2756:
2755:
2710:Signal processing
2429:Doob's upcrossing
2424:Doob's martingale
2388:Engelbert–Schmidt
2331:Donsker's theorem
2265:Feller-continuous
2133:Rendleman–Bartter
1923:Dirichlet process
1840:Branching process
1809:Telegraph process
1702:Geometric process
1682:Empirical process
1672:Diffusion process
1528:Branching process
1523:Bernoulli process
1473:
1472:
1426:Exchangeable bond
1352:Yield to maturity
1204:Exchangeable bond
1126:Subordinated debt
362:
160:
159:
125:
2791:
2779:Financial models
2730:Machine learning
2617:Usual hypotheses
2500:Girsanov theorem
2485:Dynkin's formula
2250:Continuous paths
2158:Actuarial models
2098:Garman–Kohlhagen
2068:Black–Karasinski
2063:Black–Derman–Toy
2050:Financial models
1916:Fields and other
1845:Gaussian process
1794:Sigma-martingale
1598:Additive process
1500:
1493:
1486:
1477:
1416:Convertible bond
1259:Zero-coupon bond
1199:Convertible bond
1184:Commercial paper
1067:
1060:
1053:
1044:
1033:, Andrea Ruberto
1019:
1017:
1007:
1005:
1004:
998:
991:
978:
976:
965:
952:
950:
939:
929:
923:
903:
902:
900:
899:
889:
883:
873:
867:
866:
864:
863:
854:. Archived from
848:
842:
841:
839:
838:
829:. Archived from
823:
817:
816:
814:
812:
803:
795:
789:
778:
772:
771:
769:
768:
762:
756:. Archived from
755:
747:
693:
691:
690:
685:
683:
682:
656:
655:
616:, the model is:
615:
613:
612:
607:
584:
582:
581:
576:
573:
572:
545:
543:
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537:
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401:
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379:
363:
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347:
338:
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251:binomial lattice
176:short-rate model
136:
134:
133:
128:
126:
118:
116:
115:
100:
92:
91:
82:
77:
76:
23:Step 0. Set the
16:
2799:
2798:
2794:
2793:
2792:
2790:
2789:
2788:
2759:
2758:
2757:
2752:
2734:
2695:Queueing theory
2638:
2580:Skorokhod space
2443:
2434:Kunita–Watanabe
2405:
2371:Sanov's theorem
2341:Ergodic theorem
2314:
2310:Time-reversible
2228:
2191:Queueing models
2185:
2181:Sparre–Anderson
2171:Cramér–Lundberg
2152:
2138:SABR volatility
2044:
2001:
1953:Boolean network
1911:
1897:Renewal process
1828:
1777:Non-homogeneous
1767:Poisson process
1657:Contact process
1620:Brownian motion
1590:Continuous time
1584:
1578:Maximal entropy
1509:
1504:
1474:
1469:
1440:
1431:Extendible bond
1421:Embedded option
1397:
1361:
1263:
1224:High-yield debt
1214:Fixed rate bond
1209:Extendible bond
1157:
1138:Government bond
1133:Distressed debt
1099:
1076:
1071:
1027:
1022:
1015:
1010:
1002:
1000:
996:
989:
981:
974:
963:
955:
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797:
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792:
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766:
764:
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753:
749:
748:
741:
731:
703:Newton's method
674:
647:
621:
620:
597:
596:
564:
556:
555:
548:Brownian motion
525:
520:
519:
495:
490:
489:
465:
460:
459:
437:
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407:
393:
352:
324:
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293:
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247:
212:
174:) is a popular
107:
83:
68:
54:
53:
51:
47:
28:
12:
11:
5:
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2787:
2786:
2781:
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2754:
2753:
2751:
2750:
2745:
2743:List of topics
2739:
2736:
2735:
2733:
2732:
2727:
2722:
2717:
2712:
2707:
2702:
2700:Renewal theory
2697:
2692:
2687:
2682:
2677:
2672:
2667:
2665:Ergodic theory
2662:
2657:
2655:Control theory
2652:
2646:
2644:
2640:
2639:
2637:
2636:
2635:
2634:
2629:
2619:
2614:
2609:
2604:
2599:
2598:
2597:
2587:
2585:Snell envelope
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2572:
2567:
2562:
2557:
2552:
2547:
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2537:
2532:
2527:
2522:
2517:
2512:
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2431:
2426:
2421:
2415:
2413:
2407:
2406:
2404:
2403:
2384:Borel–Cantelli
2373:
2368:
2363:
2358:
2353:
2348:
2343:
2338:
2333:
2328:
2322:
2320:
2319:Limit theorems
2316:
2315:
2313:
2312:
2307:
2302:
2297:
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2110:
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2100:
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2027:
2022:
2017:
2011:
2009:
2003:
2002:
2000:
1999:
1994:
1989:
1988:
1987:
1982:
1972:
1967:
1962:
1957:
1956:
1955:
1950:
1940:
1938:Hopfield model
1935:
1930:
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1917:
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1910:
1909:
1904:
1899:
1894:
1889:
1884:
1883:
1882:
1877:
1872:
1867:
1857:
1855:Markov process
1852:
1847:
1842:
1836:
1834:
1830:
1829:
1827:
1826:
1824:Wiener sausage
1821:
1819:Wiener process
1816:
1811:
1806:
1801:
1799:Stable process
1796:
1791:
1789:Semimartingale
1786:
1781:
1780:
1779:
1774:
1764:
1759:
1754:
1749:
1744:
1739:
1734:
1732:Jump diffusion
1729:
1724:
1719:
1714:
1709:
1707:Hawkes process
1704:
1699:
1694:
1689:
1687:Feller process
1684:
1679:
1674:
1669:
1664:
1659:
1654:
1652:Cauchy process
1649:
1648:
1647:
1642:
1637:
1632:
1627:
1617:
1616:
1615:
1605:
1603:Bessel process
1600:
1594:
1592:
1586:
1585:
1583:
1582:
1581:
1580:
1575:
1570:
1565:
1555:
1550:
1545:
1540:
1535:
1530:
1525:
1519:
1517:
1511:
1510:
1505:
1503:
1502:
1495:
1488:
1480:
1471:
1470:
1468:
1467:
1461:
1455:
1448:
1446:
1442:
1441:
1439:
1438:
1433:
1428:
1423:
1418:
1413:
1407:
1405:
1399:
1398:
1396:
1395:
1390:
1385:
1380:
1375:
1369:
1367:
1363:
1362:
1360:
1359:
1354:
1349:
1344:
1339:
1334:
1332:Risk-free bond
1329:
1324:
1319:
1317:Mortgage yield
1314:
1309:
1304:
1299:
1294:
1289:
1284:
1279:
1273:
1271:
1269:Bond valuation
1265:
1264:
1262:
1261:
1256:
1251:
1246:
1244:Perpetual bond
1241:
1236:
1231:
1226:
1221:
1216:
1211:
1206:
1201:
1196:
1191:
1186:
1181:
1176:
1171:
1165:
1163:
1159:
1158:
1156:
1155:
1150:
1148:Municipal bond
1145:
1140:
1135:
1130:
1129:
1128:
1123:
1116:Corporate bond
1113:
1107:
1105:
1101:
1100:
1098:
1097:
1092:
1087:
1081:
1078:
1077:
1072:
1070:
1069:
1062:
1055:
1047:
1041:
1040:
1034:
1026:
1025:External links
1023:
1021:
1020:
1008:
979:
977:on 2012-04-22.
953:
951:on 2008-09-10.
930:
928:: vol.7 No. 3.
908:
905:
904:
884:
868:
843:
818:
790:
773:
738:
737:
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727:
695:
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681:
677:
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669:
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631:
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322:
318:
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312:
309:
306:
303:
300:
246:
243:
220:Emanuel Derman
211:
208:
200:mean-reverting
158:
157:
153:
152:
141:
138:
124:
121:
114:
110:
106:
103:
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90:
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49:
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42:
13:
10:
9:
6:
4:
3:
2:
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2668:
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2656:
2653:
2651:
2648:
2647:
2645:
2641:
2633:
2630:
2628:
2625:
2624:
2623:
2620:
2618:
2615:
2613:
2610:
2608:
2605:
2603:
2602:Stopping time
2600:
2596:
2593:
2592:
2591:
2588:
2586:
2583:
2581:
2578:
2576:
2573:
2571:
2568:
2566:
2563:
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2558:
2556:
2553:
2551:
2548:
2546:
2543:
2541:
2538:
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2528:
2526:
2523:
2521:
2518:
2516:
2513:
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2508:
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2498:
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2468:
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2427:
2425:
2422:
2420:
2417:
2416:
2414:
2412:
2408:
2401:
2397:
2393:
2392:Hewitt–Savage
2389:
2385:
2381:
2377:
2376:Zero–one laws
2374:
2372:
2369:
2367:
2364:
2362:
2359:
2357:
2354:
2352:
2349:
2347:
2344:
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2339:
2337:
2334:
2332:
2329:
2327:
2324:
2323:
2321:
2317:
2311:
2308:
2306:
2303:
2301:
2298:
2296:
2293:
2291:
2288:
2286:
2283:
2281:
2278:
2276:
2273:
2271:
2268:
2266:
2263:
2261:
2258:
2256:
2253:
2251:
2248:
2246:
2243:
2241:
2238:
2237:
2235:
2231:
2225:
2222:
2220:
2217:
2215:
2212:
2210:
2207:
2205:
2202:
2200:
2197:
2196:
2194:
2192:
2188:
2182:
2179:
2177:
2174:
2172:
2169:
2167:
2164:
2163:
2161:
2159:
2155:
2149:
2146:
2144:
2141:
2139:
2136:
2134:
2131:
2129:
2126:
2124:
2121:
2119:
2116:
2114:
2111:
2109:
2106:
2104:
2101:
2099:
2096:
2094:
2091:
2089:
2086:
2084:
2081:
2079:
2076:
2074:
2073:Black–Scholes
2071:
2069:
2066:
2064:
2061:
2059:
2056:
2055:
2053:
2051:
2047:
2041:
2038:
2036:
2033:
2031:
2028:
2026:
2023:
2021:
2018:
2016:
2013:
2012:
2010:
2008:
2004:
1998:
1995:
1993:
1990:
1986:
1983:
1981:
1978:
1977:
1976:
1975:Point process
1973:
1971:
1968:
1966:
1963:
1961:
1958:
1954:
1951:
1949:
1946:
1945:
1944:
1941:
1939:
1936:
1934:
1933:Gibbs measure
1931:
1929:
1926:
1924:
1921:
1920:
1918:
1914:
1908:
1905:
1903:
1900:
1898:
1895:
1893:
1890:
1888:
1885:
1881:
1878:
1876:
1873:
1871:
1868:
1866:
1863:
1862:
1861:
1858:
1856:
1853:
1851:
1848:
1846:
1843:
1841:
1838:
1837:
1835:
1831:
1825:
1822:
1820:
1817:
1815:
1812:
1810:
1807:
1805:
1802:
1800:
1797:
1795:
1792:
1790:
1787:
1785:
1782:
1778:
1775:
1773:
1770:
1769:
1768:
1765:
1763:
1760:
1758:
1755:
1753:
1750:
1748:
1745:
1743:
1740:
1738:
1735:
1733:
1730:
1728:
1725:
1723:
1722:Itô diffusion
1720:
1718:
1715:
1713:
1710:
1708:
1705:
1703:
1700:
1698:
1697:Gamma process
1695:
1693:
1690:
1688:
1685:
1683:
1680:
1678:
1675:
1673:
1670:
1668:
1665:
1663:
1660:
1658:
1655:
1653:
1650:
1646:
1643:
1641:
1638:
1636:
1633:
1631:
1628:
1626:
1623:
1622:
1621:
1618:
1614:
1611:
1610:
1609:
1606:
1604:
1601:
1599:
1596:
1595:
1593:
1591:
1587:
1579:
1576:
1574:
1571:
1569:
1568:Self-avoiding
1566:
1564:
1561:
1560:
1559:
1556:
1554:
1553:Moran process
1551:
1549:
1546:
1544:
1541:
1539:
1536:
1534:
1531:
1529:
1526:
1524:
1521:
1520:
1518:
1516:
1515:Discrete time
1512:
1508:
1501:
1496:
1494:
1489:
1487:
1482:
1481:
1478:
1465:
1462:
1459:
1456:
1453:
1450:
1449:
1447:
1443:
1437:
1436:Puttable bond
1434:
1432:
1429:
1427:
1424:
1422:
1419:
1417:
1414:
1412:
1411:Callable bond
1409:
1408:
1406:
1404:
1400:
1394:
1391:
1389:
1386:
1384:
1381:
1379:
1376:
1374:
1371:
1370:
1368:
1364:
1358:
1355:
1353:
1350:
1348:
1345:
1343:
1340:
1338:
1335:
1333:
1330:
1328:
1325:
1323:
1322:Nominal yield
1320:
1318:
1315:
1313:
1310:
1308:
1305:
1303:
1300:
1298:
1297:Current yield
1295:
1293:
1292:Credit spread
1290:
1288:
1285:
1283:
1280:
1278:
1275:
1274:
1272:
1270:
1266:
1260:
1257:
1255:
1252:
1250:
1249:Puttable bond
1247:
1245:
1242:
1240:
1237:
1235:
1232:
1230:
1227:
1225:
1222:
1220:
1217:
1215:
1212:
1210:
1207:
1205:
1202:
1200:
1197:
1195:
1192:
1190:
1187:
1185:
1182:
1180:
1179:Callable bond
1177:
1175:
1172:
1170:
1167:
1166:
1164:
1160:
1154:
1151:
1149:
1146:
1144:
1141:
1139:
1136:
1134:
1131:
1127:
1124:
1122:
1119:
1118:
1117:
1114:
1112:
1109:
1108:
1106:
1102:
1096:
1093:
1091:
1088:
1086:
1083:
1082:
1079:
1075:
1068:
1063:
1061:
1056:
1054:
1049:
1048:
1045:
1038:
1035:
1032:
1029:
1028:
1024:
1014:
1009:
999:on 2011-01-29
995:
988:
984:
980:
973:
969:
962:
958:
954:
947:
943:
936:
931:
927:
920:
915:
914:
913:
912:
894:
888:
885:
881:
877:
872:
869:
858:on 2016-05-24
857:
853:
847:
844:
833:on 2010-03-28
832:
828:
822:
819:
807:
800:
794:
791:
788:
785:, p. 410, at
784:
783:
777:
774:
763:on 2011-10-07
759:
752:
746:
744:
740:
736:
735:
728:
726:
724:
720:
716:
712:
708:
707:secant method
704:
700:
679:
675:
671:
667:
664:
661:
658:
652:
648:
644:
638:
632:
629:
626:
619:
618:
617:
602:
588:
569:
565:
561:
553:
549:
546:= a standard
530:
526:
518:
500:
496:
488:
470:
466:
458:
442:
435:
432:
431:
430:
429:
412:
408:
404:
398:
394:
390:
387:
384:
380:
373:
367:
364:
357:
353:
348:
344:
340:
334:
329:
325:
320:
316:
310:
304:
301:
298:
291:
290:
289:
287:
282:
280:
276:
272:
268:
264:
260:
256:
252:
244:
242:
240:
239:
235:
231:
230:
225:
224:Goldman Sachs
221:
217:
216:Fischer Black
209:
207:
205:
201:
197:
193:
189:
185:
181:
177:
173:
169:
165:
156:
150:
146:
142:
139:
122:
112:
108:
104:
101:
97:
88:
84:
79:
73:
69:
62:
59:
43:
40:
39:
38:
36:
32:
26:
21:
18:
17:
2660:Econometrics
2622:Wiener space
2510:Itô integral
2411:Inequalities
2300:Self-similar
2270:Gauss–Markov
2260:Exchangeable
2240:Càdlàg paths
2176:Risk process
2128:LIBOR market
2062:
1997:Random graph
1992:Random field
1804:Superprocess
1742:Lévy process
1737:Jump process
1712:Hunt process
1548:Markov chain
1445:Institutions
1403:Bond options
1347:Yield spread
1239:Lottery bond
1169:Accrual bond
1095:Fixed income
1039:, Serkan Gur
1001:. Retrieved
994:the original
972:the original
967:
946:the original
941:
925:
910:
909:
896:. Retrieved
887:
876:Phelim Boyle
871:
860:. Retrieved
856:the original
846:
835:. Retrieved
831:the original
821:
809:. Retrieved
805:
793:
787:Google Books
780:
776:
765:. Retrieved
758:the original
733:
732:
696:
594:
587:differential
552:risk-neutral
283:
248:
236:
227:
213:
180:bond options
171:
167:
161:
154:
22:
19:
2705:Ruin theory
2643:Disciplines
2515:Itô's lemma
2290:Predictable
1965:Percolation
1948:Potts model
1943:Ising model
1907:White noise
1865:Differences
1727:Itô process
1667:Cox process
1563:Loop-erased
1558:Random walk
1342:Yield curve
1302:Dirty price
1277:Clean price
1153:Global bond
1121:Senior debt
1111:Agency bond
1074:Bond market
723:martingales
715:algorithmic
261:), and the
259:yield curve
35:iteratively
2763:Categories
2715:Statistics
2495:Filtration
2396:Kolmogorov
2380:Blumenthal
2305:Stationary
2245:Continuous
2233:Properties
2118:Hull–White
1860:Martingale
1747:Local time
1635:Fractional
1613:pure birth
1003:2011-04-08
898:2021-06-09
862:2010-06-14
837:2010-04-26
767:2011-07-21
729:References
271:as implied
255:calibrates
196:stochastic
186:and other
145:zero-price
2627:Classical
1640:Geometric
1630:Excursion
1282:Convexity
1090:Debenture
957:Boyle, P.
711:bisection
701:—such as
668:σ
649:θ
633:
603:σ
497:σ
467:θ
395:σ
368:
354:σ
341:σ
326:θ
305:
269:(usually
184:swaptions
120:Δ
109:σ
63:
31:spot rate
2748:Category
2632:Abstract
2166:Bühlmann
1772:Compound
1357:Z-spread
1312:I-spread
1307:Duration
985:(2008).
983:Hull, J.
911:Articles
811:19 March
550:under a
349:′
275:Black-76
245:Formulae
2255:Ergodic
2143:Vašíček
1985:Poisson
1645:Meander
1466:(SIFMA)
806:soa.org
273:by the
210:History
2595:Tanaka
2280:Mixing
2275:Markov
2148:Wilkie
2113:Ho–Lee
2108:Heston
1880:Super-
1625:Bridge
1573:Biased
1460:(ICMA)
1454:(CMSA)
1287:Coupon
1189:Consol
433:where,
253:, one
234:memoir
190:; see
166:, the
2448:Tools
2224:M/M/c
2219:M/M/1
2214:M/G/1
2204:Fluid
1870:Local
1016:(PDF)
997:(PDF)
990:(PDF)
975:(PDF)
964:(PDF)
949:(PDF)
938:(PDF)
922:(PDF)
802:(PDF)
761:(PDF)
754:(PDF)
734:Notes
709:) or
705:(the
2400:Lévy
2199:Bulk
2083:Chen
1875:Sub-
1833:Both
1085:Bond
813:2024
585:its
265:for
1980:Cox
721:or
172:BDT
162:In
48:; r
2765::
2398:,
2394:,
2390:,
2386:,
2382:,
966:.
940:.
924:.
804:.
742:^
725:.
630:ln
365:ln
302:ln
288::
281:.
241:.
218:,
182:,
60:ln
37::
33:,
2402:)
2378:(
1499:e
1492:t
1485:v
1066:e
1059:t
1052:v
1006:.
901:.
865:.
840:.
815:.
770:.
680:t
676:W
672:d
665:+
662:t
659:d
653:t
645:=
642:)
639:r
636:(
627:d
589:.
570:t
566:W
562:d
531:t
527:W
501:t
471:t
443:r
413:t
409:W
405:d
399:t
391:+
388:t
385:d
381:]
377:)
374:r
371:(
358:t
345:t
335:+
330:t
321:[
317:=
314:)
311:r
308:(
299:d
170:(
123:t
113:i
105:=
102:2
98:/
94:)
89:d
85:r
80:/
74:u
70:r
66:(
50:d
46:u
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.